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$\begingroup$Can the first digit of the answer be a zero? If that's the case, I've found a solution,$\endgroup$
– астон вілла олоф мэллбэргFeb 17 '17 at 1:05

$\begingroup$астон вілла олоф мэллбэрг: Normally not in puzzles like this, but there are exceptions. In fact, I start with the thousands digit of the sum being $1$$\endgroup$
– Ross MillikanFeb 17 '17 at 1:06

1

$\begingroup$A more usual way to write this is $$\ \ ABC\\ \underline {+DEF}\\GHJK$$ which gives a way to refer to the digits. My previous comment then could include $G=1$ because that is all you can carry from the addition of two numbers.$\endgroup$
– Ross MillikanFeb 17 '17 at 1:13

$\begingroup$Hmm I count 96 lines (commutativity is not duplication) ... and that fits the output of my program :-) Btw I think OP looks for a mathematical resolution, rather than a raw output ... Or at least show your algorithm! (well, I could have provided mine as well, but this is not stackoverflow, so not sure it qualifies!)$\endgroup$
– ringøFeb 17 '17 at 2:00

$\begingroup$@OscarLanzi well, 96 out of 10 billions possibilities is not so much...$\endgroup$
– ringøFeb 17 '17 at 2:05

$\begingroup$@ringo not $10^{10}$. All digits must be distinct which removes most of the tuples; digit G has to be $1$; the other initial digits cannot be zero; and in my book two solutions that differ only by switching the addends are not really different. Thus I am left with only about $130,000$ candidates.$\endgroup$
– Oscar LanziFeb 17 '17 at 2:38